Question

Alice and Bob Ray use cabbages, tomatoes and onions to make two kinds of relish: chow-chow and tomato. A jar of Chow-chow contains 8 ounces of cabbage, 3 ounces of tomatoes, and 3 ounces of onions. A jar of tomato relish contains 6 ounces of tomatoes, 6 ounces of cabbage, and 2 ounces of onions. The Rays have 120 ounces of cabbages, 90 ounces of tomatoes and 45 ounces of onions each week. The profit for a jar of chow-chow is $2.25 and the profit for a jar of tomato relish is $1.95. The Rays want to know how many jars of each kind of relish to make each week to maximize the profit. What is the shadow price for a tomato?

Answer 1

Data
	Cabbage	Tomato	Onion	Profit
1 jar chow-chow relish	8	3	3	2.25
1 jar tomato relish	6	6	2	1.95
Availability	120	90	45

This is a maximization problem which can be solved by Linear Programming.

Formulating Linear Programming:

1. Define Decision Variables: Let C and T be the no. of jars of Chow Chow Relish and Tomato Relish made each week respectively.

2. Objective Function: The objective is to maximize profit.

Zmax = 2.25C+1.95T

3. Defining Constraints:

8C+6T ≤ 120 (Cabbage Availability Constraint)

3C+6T ≤ 90 (Tomatoes Availability Constraint)

3C+2T ≤ 45 (Onion Availability Constraint)

Explanation: 1 Jar or Chow Chow and Tomato Relish contain 8 ounces and 6 ounces of Cabbage respectively.

C jars of Chow Chow would contain 8C ounces of cabbage and T jars of Tomato Relish would contain 6T ounces of Cabbage. The total cabbage required for making C and T jars of relishes should not be more than 120 ounces.

4. Solving the LP in solver:

Excel Formulae used:

a. Solver Parameters:

Enter the details in the excel sheet as shown in the screenshot and then add the required parameters which is also shown in the screenshot.

After the parameters are added click on solve to get the solution in the changing cells and objective cell. This will also pop up a dialogue box, where you can select sensitivity and click ok. This generates the sensityvity report which will provide details about the shadow price.

2. Solution :

The number of jars they should make each week

	Chow chow	Tomato
No. of Jars	6	12

Optimal profit = 36.9

In the sensitivity report generated, check for the shadow price of constraint leveled as tomato.

Please ignore the objective part in every name.

The shadow price of the tomato is 0.07. This means for every unit change in the Constraint R.H side of tomato (90), the objective or optimal value will change by 0.07 units.